441 research outputs found

    Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

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    A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

    Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

    Get PDF
    A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

    Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

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    This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial L1×L2×L3L_1\times L_2\times L_3 lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L1×L2×L3L_1\times L_2\times L_3 lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L×1×1L\times 1\times 1 lattice chain in a 3D rectangular "tube" with LL up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large LL.Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1408.383

    Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

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    This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let Ik:=(−π,π)kI_k:=(-\pi,\pi)^k, let Ms\mathcal M_s be the linear space of complex s×ss\times s matrices, and let f,g:Ik→Msf,g:I_k\to\mathcal M_s be functions whose components fij, gij:Ik→C, i,j=1,…,s,f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s, belong to L∞L^\infty. Consider the matrices Tn−1(g)Tn(f)T_n^{-1}(g)T_n(f), where n:=(n1,…,nk)n:=(n_1,\ldots,n_k) varies in Nk\mathbb N^k and Tn(f),Tn(g)T_n(f),T_n(g) are the multilevel block Toeplitz matrices of size n1⋯nksn_1\cdots n_ks generated by f,gf,g. Then {Tn−1(g)Tn(f)}n∈Nk∼λg−1f\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f, i.e. the family of matrices {Tn−1(g)Tn(f)}n∈Nk\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k} has a global (asymptotic) spectral distribution described by the function g−1fg^{-1}f, provided gg possesses certain properties (which ensure in particular the invertibility of Tn−1(g)T_n^{-1}(g) for all nn) and the following topological conditions are met: the essential range of g−1fg^{-1}f, defined as the union of the essential ranges of the eigenvalue functions λj(g−1f), j=1,…,s\lambda_j(g^{-1}f),\ j=1,\ldots,s, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case g=1g=1. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure
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